1952 Polish MO Finals

Part of Polish MO Finals

Subcontests

(5)

1

spiral staircase in circular tower

In a circular tower with an internal diameter of

2

m, there is a spiral staircase with a height of

6

m. The height of each stair step is

0.15

m. In the horizontal projection, the steps form adjacent circular sections with an angle of

18^\circ

. The narrower ends of the steps are mounted in a round pillar with a diameter of

0.64

m, the axis of which coincides with the axis of the tower. Calculate the greatest length of a straight rod that can be moved up these stairs from the bottom to the top (do not take into account the thickness of the rod or the thickness of the boards from which the stairs are made).

1

last digit of 1 + 2 + 3 + ...+ n,

Prove that none of the digits

2

4

7

9

can be the last digit of a number

1 + 2 + 3 + \ldots + n,

n

is a natural number.

1

cos 3A + cos 3B + cos 3C = 1

Prove that if the angles

A

B

C

of a triangle satisfy the equation

\cos 3A + \cos 3B + \cos 3C = 1,

then one of these angles equals

120^\circ

1

quad construction

Construct the quadrilateral

ABCD

given the lengths of the sides

AB

CD

and the angles of the quadrilateral.

1

triangle on sides of triangle, equal ratio

BC

CA

AB

of the triangle

ABC

M

N

P

are taken, respectively, in such a way that

\frac{BM}{MC} = \frac{CN}{NA} = \frac{AP}{PB} = k,

k

means a given number greater than

1

, then the segments

AM

BN

CP

were drawn . Given the area

S

of the triangle

ABC

, calculate the area of the triangle bounded by the lines

AM

BN

CP